Abstract—The objective of this study is to investigate the

static analysis of symmetrically and anti-symmetrically

laminated elastic orthotropic composite circular rods with

rectangular cross-section under the influence of distributed

forces based on the mixed finite element method with

Timoshenko theory. The Poisson’s ratios are incorporated in

the constitutive relation.

Index Terms— Circular rods, Composite Material, Mixed

Finite Elements, Static Analysis

I. INTRODUCTION

HE increased use of composites in many applications

such as aerospace, biomedical, mechanical, aircrafts,

marine industry, robot arms, etc. due to their attractive

properties in strength, stiffness and lightness has resulted in

a growing demand for engineers in the design of laminated

structures made of fiber-reinforced composite materials.

Numerical texts dealing with the mechanics of composites

by using various beam theories have been published to

satisfy this demand [1]-[11]. Although the static analysis of

laminated elastic isotropic and transversely isotropic

composite rods is investigated intensively in the literature,

studies with orthotropic lay-ups are quite rare.

In order to achieve maximum structural efficiency,

designing laminates made of laminae that have different ply

angles is a necessity. In this study, the Poisson’s ratios are

incorporated in the constitutive relations not to cause loss of

some stiffness coefficients for angle-ply laminae. The

constitutive equations of generally layered orthotropic rods

are derived by reducing the constitutive relations of

orthotropic materials for three-dimensional body [12].

Mixed finite element method with Timoshenko beam theory

is used. The determination of the functional is based on the

procedure given by [13] using Gâteaux differential [14].

II. CONSTITUTIVE RELATIONS OF COMPOSITE RODS

A. Constitutive Relations for a Single Lamina

The generalized Hooke’s laws for elasticity and

compliance matrices are defined as,

: or :E C (1)

where, , , E and 1( )C E denotes the stress tensor,

strain tensor, elasticity matrix and compliance matrix,

respectively. The constitutive relations of a transversely

Manuscript received March 02, 2016; revised April 06, 2016.

All authors are with Istanbul Technical University, 34469, Istanbul,

TURKEY Email of the corresponding author : umitaribas@hotmail.com

orthotropic material have different material properties in

three mutually perpendicular directions at a point of the

body and further perpendicular planes of material property

symmetry. Thus,

11 12 131 1

12 22 232 2

13 23 333 3

4423 23

5531 31

6612 12

0 0 0

0 0 0

0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

E E E

E E E

E E E

E

E

E

(2)

Each layer in a laminate can have principal material

direction orientations via versus to global laminate axes to

meet the particular requirements of stiffness and strength.

Thus, the transformation between the principle axes of the

laminae and the global axes of the laminate must be

considered.

1 T: : : :T E T E (3)

where, T is the transformation matrix.

11 12 13 16

12 22 23 26

13 23 33 36

44 45

45 55

16 26 36 66

0 0

0 0

0 0

0 0 0 0

0 0 0 0

0 0

E E E E

E E E E

E E E E

E E

E E

E E E E

E = (4)

More detailed information about the components of

elasticity matrices in Eqns. (2) and (4), and the

transformation matrix for a rotation about b-axis (Fig 1.)

exists in [15].

Fig. 1. Frenet coordinates of the circular rod

with a rectangular cross-section.

The constitutive equations for a single lamina are derived

by reducing the constitutive equations of orthotropic

materials for three-dimensional body. The assumptions

made on stresses, in accordance with the classical rod

theory, are as follows [16],

0n b nb (5)

where, the stress components in Frenet coordinates via

versus in a three dimensional body are,

Static Analysis of Moderately Thick,

Composite, Circular Rods via Mixed FEM

Ü.N. Arıbaş, A. Kutlu, and M.H. Omurtag

T

Proceedings of the World Congress on Engineering 2016 Vol II WCE 2016, June 29 - July 1, 2016, London, U.K.

ISBN: 978-988-14048-0-0 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)

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1 2 3

23 31 12

, ,

, ,

t n b

nb bt tn

(6)

By paying attention to Eqns. (1), (4) and (5), the strain

components b , n , nb are calculated with respect to the

t , bt and tn for the rods and inserted into the equalities

[12],

11 13

22

13 33

0

0 0

0

t t

bt bt

tn tn

(7)

The displacements of an arbitrary point on the cross-

section for a circular rod (Fig. 1) are stated as,

*

*

*

t t n b

n n t

b b t

u u b n

u u b

u u n

(8)

where, *

tu , *

nu , *

bu are displacements at the rod continuum

and tu , nu , bu are displacements on the rod axis and t ,

n and b present the rotations of the rod on the t, n and b

Frenet Coordinates, respectively. The strains can be derived

from Eqn. (8) as [17],

0

0

btn

t

tt b

bt

tn tt n

u

ttt

u ub n

b t t

u utn t

(9)

B. Lamination

A lamina is a flat arrangement of the laminate. When a

stack of laminae with various orientations of principle

material directions are bonded together and act as an integral

structural element to match the loading environment by

tailoring the directional dependence of strength and stiffness

of the structural element and get corrosion resistance,

acoustical insulation and temperature-dependent behavior,

etc. is called laminate. It is assumed that the strain

distribution is linear through the laminate thickness and no

slip occurs between the boundaries of laminae but the stress

distribution has discontinuities at boundaries between

laminae. Thus, the forces and moments for a laminate can be

derived by analytical integration of the stresses in each

lamina through the thickness of the laminate.

Fig. 2. The stresses with respect to the Frenet Coordinates,

N: total number of layers.

Fig. 3. The moments with respect to the Frenet Coordinates.

Namely; Derivation of forces,

1

0.5

0.51

d dL L

L L

N n b

t tn b

L

T b n

(10)

1

0.5

0.51

d dL L

L L

N n b

b btn b

L

T b n

(11)

1

0.5

0.51

d dL L

L L

N n b

n tnn b

L

T b n

(12)

Derivation of moments,

1

1

0.5

0.51

0.5

0.51

d d

d d

L L

L L

LL

L

L

N n b

t tnn b

L

bN n

tbn

L b

M b b n

n n b

(13)

1

0.5

0.51

d dL L

L L

N n b

n tn b

L

M b b n

(14)

1

0.5

0.51

d dL L

L L

N b n

b tb n

L

M n n b

(15)

where, N is the number of the laminae in a laminate, Ln is

the width of the layer and Lb and 1Lb are the directed

distances to the bottom and the top of the thL layer where b

is positive upward. Finally, the constitutive relations are

obtained by using Eqns. (10)-(15) in a matrix form:

1

t

t n

t

n t bL LNT TMb

L LtL MT Mt

n

nb

b

u

t

u uT n t

T u u

T b t

M

tM

M

t

t

E E

E E (16)

L

TME and L

MTE are coupling matrices. Also please note that

TL L

TM MTE E . The compliance matrix of a laminate is

obtained by 1C = E .

Proceedings of the World Congress on Engineering 2016 Vol II WCE 2016, June 29 - July 1, 2016, London, U.K.

ISBN: 978-988-14048-0-0 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)

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C. Functional

The field equations based on Timoshenko beam theory

for a curved rod can be given as [18].

Equilibrium Equations;

d0

d

d0

d

As

s

Tq + u

Mt T m + I

(17)

Kinematic Equations;

d0

d

d0

d

MT M

T TM

s

s

C T C M

uC T C M t

(18)

In general sense, the field equations can be written in an

operator form Q = Ly - f and if the operator is potential, the

equality * *d , , d , ,Q y y y Q y y y must be satisfied

[18]. d ,Q y y and *d ,Q y y are Gâteaux derivatives of

the operator in directions of y and *y . After proving the

operator to be potential, the functional including the rotary

inertia and shear influence is obtained as follows:

σ σ

d d(y) , ,

ds ds

1x , ,

2

ˆ, , ,

ˆ ˆ ˆ, , ,

T TM

MT M

I

T Mu q u m

t T C T T C M T

C T M C M M T u

M T T u M M

(19)

where, q is distributed external forces, m is distributed

external moments along the rod axis, TMC and

MTC are

coupling compliance matrices, u are displacements on the

rod axis, are rotations of the rod, T are forces and M

are moments of cross-section and quantities with hat are

known values on the boundary.

D. The Mixed Finite Element Algorithm

The linear shape functions ( ) /i j and

( ) /j i are employed in the finite element

formulation, where ( )j i . ,i j represent the node

numbers of the curved element. The field variables at a node

are,

{ , , , , , , , , , , , }Tt n b t n b t n b t n bu u u T T T M M M X (20)

The curvatures are satisfied exactly at the nodal points

and linearly interpolated through the element [19].

III. NUMERICAL EXAMPLES

Numerical results are verified by ANSYS 14.5 for the

static analysis of a straight rod with three orthotropic lay-

ups. Static analysis of symmetrically and anti-symmetrically

layered cross-ply circular rod with four orthotropic lay-ups

is investigated for three different lay-up conditions as

benchmark example.

Orthotropic material properties used for convergence and

benchmark examples are given as [16] in Table I.

TABLE I

MATERIAL PROPERTIES OF ORTHOTROPIC MATERIAL

Young's Modulus

(N/m2) x109

Shear Modulus

(N/m2) x109

Poisson's

Ratio

Et = 39.3 Gtn = 29.3 νtn = 0.613

En = 152 Gtb = 1.6 νtb = 0.265

Eb = 22.9 Gnb = 4.1 νnb = 0.209

A. Convergence Analysis

Static analysis of symmetrically laminated elastic straight

beam with three orthotropic cross-ply layers under the

influence of distributed force is investigated. Both ends of

the beam have fixed supports and 10 N/m distributed load

applied along the length of the beam (18m). Each layer has

the same orthotropic material properties given in Table I

with different orientations (0°,90°,0°). Dimensions of the

thickness and width of layers are given in Figure 4.

Fig. 4. Cross-section of the convergence example.

Results are obtained by ANSYS 14.5 with three bend

solid layer elements fixed at supports, distributed load is

applied at the top surface of top layer, core layer's

orientation is given by applying a local Cartesian Coordinate

system for core layer. A convergence analysis is performed

for 20, 40, 60, 80 and 100 elements by using this study in

order to attain the necessary precision for the results (Table

II).

TABLE II

CONVERGENCE VIA VERSUS ANSYS

REACTION

MAX. DISP.

(m)×10-5

FORCE

(N)

MOMENT

(Nm)

Nu

mb

er o

f

elem

ents

in

th

is

stu

dy

20 9.662 90 267.03

40 9.569 90 269.33

60 9.552 90 269.70

80 9.546 90 269.83

100 9.543 90 269.89

ANSYS 9.448 90 270.14

It is observed that the result are quite satisfactory, the

reaction forces present better convergence than

displacements. The maximum difference is obtained 1% for

60 elements compare to ANSYS solid elements.

Proceedings of the World Congress on Engineering 2016 Vol II WCE 2016, June 29 - July 1, 2016, London, U.K.

ISBN: 978-988-14048-0-0 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)

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B. Benchmark Examples

Static analysis of symmetrically and anti-symmetrically

laminated elastic circular rod with four orthotropic cross-ply

layers under the influence of distributed load are

investigated. Both ends of the rod have fixed support, rod

geometry is a half circle with 18m radius and 1 N/m out of

plane distributed load applied along the length of the beam

(Fig. 5).

Fig. 5. Geometry of the rod for benchmark examples.

Each layer has the same orthotropic material properties

given in Table I with different orientations (0°,90°,90°,0°),

(90°,0°,0°,90°) and (0°,90°,0°,90°). Dimensions of the

thickness and width of layers are given in Figure 6.

Fig. 6. Cross-section of benchmark examples.

Results given in Table III are obtained by using 40 finite

elements for symmetric lamination, second and third layers

are oriented 90° about b axis (I), top and bottom layers are

oriented 90° about b axis (II) and anti-symmetric

lamination, second and top layers are oriented 90° about b

axis (III).

TABLE III

SYMMETRY AND ANTI-SYMMETRY CONDITION

(I) (II) (III)

Max. Disp. (m) ×10-5 34.12 31.40 33.05

RE

AC

TIO

N

Force (N) 28.27 28.27 28.27

Mt (Nm) 95.30 95.01 95.10

Mn (Nm) 325.56 326.06 325.90

It is observed from Table III, layer-wise material

placement has great importance to reduce the maximum

displacements while considering orthotropic layered

laminated rods. Minimum displacement is obtained for

symmetric condition (II), but anti-symmetric condition (III)

presents lesser displacements compare to symmetric

condition (II).

IV. CONCLUSION

In this study, the static analysis of symmetrically and anti-

symmetrically laminated orthotropic circular rods under the

influence of distributed forces are investigated. Convergence

analysis of laminated straight rods made of orthotropic

material have been done for various finite elements and the

result are compared with ANSYS 14.5 due to the studies

with orthotropic lay-ups are quite rare in the literature.

Satisfactory results are obtained for constitutive equations of

this study with 60 finite elements. As a benchmark example,

symmetry condition is investigated for a circular rod with

four orthotropic layers. It is observed that, in composite

materials, layer-wise material placement along the thickness

direction has great importance on the response the structure.

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Proceedings of the World Congress on Engineering 2016 Vol II WCE 2016, June 29 - July 1, 2016, London, U.K.

ISBN: 978-988-14048-0-0 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)

WCE 2016